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A method for designing an integrated computational element (ICE) includes
generating an array of discrete data points and plotting the discrete
data points across a predetermined wavelength region. A line shape is
then generated that connects to and is constrained by the array of
discrete data points, and thereby generates a first transmission
function. The discrete data points are then iteratively modified based on
one or more performance criteria to generate a second transmission
function. A model transmission function corresponding to a model ICE
design is then fitted to the second transmission function to identifying
a predictive ICE design configured to detect a desired characteristic of
interest.

1. A method for designing an integrated computational element (ICE),
comprising: generating an array of discrete data points and plotting the
discrete data points across a predetermined wavelength region; generating
a line shape that connects to and is constrained by the array of discrete
data points, thereby generating a first transmission function;
iteratively modifying the discrete data points based on one or more
performance criteria to generate a second transmission function; and
fitting a model transmission function corresponding to a model ICE design
to the second transmission function, thereby identifying a predictive ICE
design configured to detect a desired characteristic of interest.

2. The method of claim 1, wherein the predetermined wavelength region
corresponds to a wavelength range wherein the desired characteristic of
interest is detectable.

3. The method of claim 1, wherein plotting the discrete data points
across the predetermined wavelength region further comprises assigning a
transmittance value to each discrete data point between zero and 1.

4. The method of claim 1, further comprising generating the array of
discrete data points using a computer-operated random number generator.

5. The method of claim 4, further comprising randomly assigning a
transmittance value to each discrete data point between zero and 1 with
the random number generator.

6. The method of claim 1, wherein generating the array of discrete data
points comprises: calculating a predetermined regression vector
corresponding to the characteristic of interest; and selecting critical
point values from the predetermined regression vector, wherein the
critical point values are used as the discrete data points.

7. The method of claim 1, further comprising generating one or both of
the first and second transmission functions using a computer-operated
point-by-point line interpolant process.

8. The method of claim 1, wherein iteratively modifying the discrete data
points based on one or more performance criteria comprises at least one
of: determining a standard error of calibration of the second
transmission function in view of the desired characteristic of interest;
and determining an output sensitivity of the second transmission function
in view of the desired characteristic of interest.

9. The method of claim 1, wherein iteratively modifying the discrete data
points comprises at least one of: iteratively altering a transmittance
value of each discrete data point to optimize the one or more performance
criteria in view of the desired characteristic of interest; and
iteratively altering a location of each discrete data point along the
predetermined wavelength region to optimize the one or more performance
criteria in view of the desired characteristic of interest.

10. The method of claim 1, wherein fitting the model transmission
function corresponding to the model ICE design to the second transmission
function comprises: generating with a computer the model ICE design
having at least one of a random number of layers and a random thickness
for each layer; iteratively modifying the model ICE design until the
model transmission function aligns with the second transmission function;
and identifying the predictive ICE design once the model transmission
function aligns with the second transmission function.

11. The method of claim 10, wherein iteratively modifying the model ICE
design comprises at least one of varying the thickness of one or more of
the layers and varying the number of layers.

12. The method of claim 1, further comprising: fabricating an ICE based
on the predictive ICE design; and using the ICE in conjunction with an
optical computing device to monitor a substance for a concentration of
the characteristic of interest.

13. A non-transitory, computer readable medium programmed with computer
executable instructions that, when executed by a processor of a computer
unit, perform the method of: generating an array of discrete data points
and plotting the discrete data points across a predetermined wavelength
region; generating a line shape that connects to and is constrained by
the array of discrete data points, thereby generating a first
transmission function; iteratively modifying the discrete data points
based on one or more performance criteria to generate a second
transmission function; and fitting a model transmission function
corresponding to a model ICE design to the second transmission function,
thereby identifying a predictive ICE design configured to detect a
desired characteristic of interest.

14. The non-transitory, computer readable medium of claim 13, wherein
plotting the discrete data points across the predetermined wavelength
region further comprises assigning a transmittance value to each discrete
data point between zero and 1.

15. The non-transitory, computer readable medium of claim 13, further
comprising generating the array of discrete data points using a
computer-operated random number generator.

16. The non-transitory, computer readable medium of claim 13, wherein
generating the array of discrete data points comprises: calculating a
predetermined regression vector corresponding to the characteristic of
interest; and selecting critical point values from the predetermined
regression vector, wherein the critical point values are used as the
discrete data points.

17. The non-transitory, computer readable medium of claim 13, further
comprising generating one or both of the first and second transmission
functions using a computer-operated point-by-point line interpolant
process.

18. The non-transitory, computer readable medium of claim 13, wherein
iteratively modifying the discrete data points based on one or more
performance criteria comprises at least one of: determining a standard
error of calibration of the second transmission function in view of the
desired characteristic of interest; and determining an output sensitivity
of the second transmission function in view of the desired characteristic
of interest.

19. The non-transitory, computer readable medium of claim 13, wherein
iteratively modifying the discrete data points comprises at least one of:
iteratively altering a transmittance value of each discrete data point to
optimize the one or more performance criteria in view of the desired
characteristic of interest; and iteratively altering a location of each
discrete data point along the predetermined wavelength region to optimize
the one or more performance criteria in view of the desired
characteristic of interest.

20. The non-transitory, computer readable medium of claim 13, further
comprising fabricating an ICE based on the predictive ICE design.

Description

BACKGROUND

[0001] Optical computing devices, also commonly referred to as
"opticoanalytical devices," can be used to analyze and monitor a sample
substance in real time. Such optical computing devices will often employ
a light source that emits electromagnetic radiation that reflects from or
is transmitted through the sample and optically interacts with an optical
processing element to determine quantitative and/or qualitative values of
one or more physical or chemical properties of the substance being
analyzed. The optical processing element may be, for example, an
integrated computational element ("ICE"). One type of ICE is an optical
thin film interference device, also known as a multivariate optical
element ("MOE"). Each ICE can be designed to operate over a continuum of
wavelengths in the electromagnetic spectrum from the vacuum-UV to
infrared (IR) ranges, or any sub-set of that region. Electromagnetic
radiation that optically interacts with the sample substance is changed
and processed by the ICE so as to be measured by a detector. The output
of the detector can be correlated to a physical or chemical property of
the substance being analyzed.

[0002] A traditional ICE includes first and second pluralities of optical
thin film layers consisting of various materials whose index of
refraction and size (e.g., thickness) varies between each layer. An ICE
design refers to the substrate, number and thickness of the respective
layers of the ICE, and the complex refractive indices of the layers. The
complex refractive index includes both the real `n` and imaginary `k`
components of the refractive index. The layers are strategically
deposited and sized so as to selectively pass predetermined fractions of
electromagnetic radiation at different wavelengths configured to
substantially mimic a regression vector corresponding to a particular
physical or chemical property of interest of a substance of interest.
Accordingly, an ICE design will exhibit a transmission function
(spectrum) that is weighted with respect to wavelength. As a result, the
output light intensity from the ICE conveyed to the detector may be
related to the physical or chemical property of interest for the
substance.

BRIEF DESCRIPTION OF THE DRAWINGS

[0003] The following figures are included to illustrate certain aspects of
the present disclosure, and should not be viewed as exclusive
embodiments. The subject matter disclosed is capable of considerable
modifications, alterations, combinations, and equivalents in form and
function, without departing from the scope of this disclosure.

[0004] FIG. 1 is a cross-sectional view of an exemplary integrated
computational element.

[0005] FIG. 2 is a schematic flowchart of an exemplary method of designing
an integrated computational element using a reverse design process.

[0008] FIG. 5 depicts the plot of FIG. 3 showing the first transmission
function in contrast with a second transmission function resulting from
modifying the discrete data points based on one or more performance
criteria.

[0010] The present invention relates to optical processing elements and,
more particularly, to improved techniques for the design of optical
processing elements for use in optical computing devices.

[0011] The present disclosure expands the design palate for optical
processing elements, such as Integrated computational elements ("ICEs"),
for use in optical computing devices. According to the improved methods
described herein, an array of discrete data points are generated and
plotted across a predetermined wavelength region. A line shape is then
generated and is constrained by the array of discrete data points, which
results in the generation of a first transmission function. The discrete
data points may then be iteratively modified based on one or more
performance criteria to generate a second transmission function. A model
transmission function corresponding to a model ICE design may then be
fitted to the second transmission function to identify a predictive ICE
design configured to detect a desired characteristic of interest.

[0012] The methods disclosed herein may prove advantageous in the design,
evaluation, and fabrication of optical processing elements (e.g., ICEs)
that may be used in the oil and gas industry, such as for monitoring and
detecting oil/gas-related substances (e.g., hydrocarbons, drilling
fluids, completion fluids, treatment fluids, etc.). The ICEs designed
using the methods disclosed herein may equally be used in other
technology fields including, but not limited to, the food industry, the
paint industry, the mining industry, the agricultural industry, the
medical and pharmaceutical industries, the automotive industry, the
cosmetics industry, water treatment facilities, and any other field where
it may be desired to monitor substances in real time.

[0013] As used herein, the term "characteristic" or "characteristic of
interest" refers to a chemical, mechanical, or physical property of a
substance to be analyzed or a sample of the substance. The characteristic
of a substance may include a quantitative or qualitative value of one or
more chemical constituents or compounds present therein or any physical
property associated therewith. Such chemical constituents and compounds
may be referred to herein as "analytes". Illustrative characteristics of
a substance that can be analyzed with the help of the optical processing
elements described herein can include, for example, chemical composition
(e.g., identity and concentration in total or of individual components),
phase presence (e.g., gas, oil, water, etc.), impurity content, pH,
alkalinity, viscosity, density, ionic strength, total dissolved solids,
salt content (e.g., salinity), porosity, opacity, bacteria content, total
hardness, transmittance, state of matter (solid, liquid, gas, emulsion,
mixtures thereof, etc.), and the like.

[0015] As used herein, the term "optically interact" or variations thereof
refers to the reflection, transmission, scattering, diffraction, or
absorption of electromagnetic radiation either on, through, or from an
optical processing element (e.g., an integrated computational element) or
a substance being analyzed with the help of the optical processing
element. Accordingly, optically interacted light refers to
electromagnetic radiation that has been reflected, transmitted,
scattered, diffracted, or absorbed by, emitted, or re-radiated, for
example, using an optical processing element, but may also apply to
optical interaction with a substance.

[0016] As used herein, the term "optical computing device" refers to an
optical device that is configured to receive an input of electromagnetic
radiation associated with a substance and produce an output of
electromagnetic radiation from an optical processing element arranged
within or otherwise associated with the optical computing device. The
optical processing element may be, for example, an integrated
computational element (ICE). The electromagnetic radiation that optically
interacts with the optical processing element is changed so as to be
readable by a detector, such that an output of the detector can be
correlated to a particular characteristic of the substance being
analyzed. The output of electromagnetic radiation from the optical
processing element can be reflected, transmitted, and/or dispersed
electromagnetic radiation. Whether the detector analyzes reflected,
transmitted, or dispersed electromagnetic radiation may be dictated by
the structural parameters of the optical computing device as well as
other considerations known to those skilled in the art.

[0017] As Indicated above, the present disclosure provides or otherwise
describes improved methods for designing optical processing elements,
such as ICEs, for use in optical computing devices. In operation, an ICE
is capable of distinguishing electromagnetic radiation related to a
characteristic of interest of a substance from electromagnetic radiation
related to other components of the substance.

[0018] FIG. 1 is a cross-sectional view of an exemplary ICE 100. As
illustrated, the ICE 100 includes a plurality of alternating thin film
layers shown as layers 102 and 104. The first layers 102 are made of a
material that exhibits a high index of refraction, such as silicon (Si),
and the second layers 104 are made of a material that exhibits a low
index of refraction, such as quartz (SiO.sub.2). Other examples of
materials that might be used include, but are not limited to, niobia and
niobium, germanium and germania, MgF, SiO, and other high and low index
materials generally known in the art. The layers 102, 104 are
strategically deposited on an optical substrate 106, such as BK-7 optical
glass. In other embodiments, the substrate 106 may be another type of
optical substrate, such as another optical glass, silica, sapphire,
silicon, germanium, zinc selenide, zinc sulfide, or various plastics such
as polycarbonate, polymethylmethacrylate (PMMA), polyvinylchloride (PVC),
diamond, ceramics, combinations thereof, and the like.

[0019] At the opposite end (e.g., opposite the substrate 106 in FIG. 1),
the ICE 100 may include a layer 108 that is generally exposed to the
environment of the device or installation. The number of layers 102, 104
and the thickness of each layer 102, 104 are determined from the spectral
attributes acquired from a spectroscopic analysis of a characteristic of
the substance being analyzed using a conventional spectroscopic
instrument. The spectrum of interest of a given characteristic typically
includes any number of different wavelengths.

[0020] It should be understood that the ICE 100 depicted in FIG. 1 does
not in fact represent an ICE configured to detect any specific
characteristic of a given substance, but is provided for purposes of
illustration only. Consequently, the number of layers 102, 104 and their
relative thicknesses, as shown in FIG. 1, bear no correlation to any
particular substance or characteristic thereof. Nor are the layers 102,
104 and their relative thicknesses necessarily drawn to scale, and
therefore should not be considered limiting of the present disclosure.

[0021] In some embodiments, the material of each layer 102, 104 can be
doped or two or more materials can be combined in a manner to achieve the
desired optical characteristic. In addition to solids, the exemplary ICE
100 may also contain liquids and/or gases, optionally in combination with
solids, in order to produce a desired optical characteristic. In the case
of gases and liquids, the ICE 100 can contain a corresponding vessel (not
shown), which houses the gases or liquids. Exemplary variations of the
ICE 100 may also include holographic optical elements, gratings,
piezoelectric, light pipe, and/or acousto-optic elements, for example,
that can create transmission, reflection, and/or absorptive properties of
interest.

[0022] The multiple layers 102, 104 exhibit different complex refractive
indices, where the complex refractive index includes both real `n` and
imaginary `k` components of the refractive index. By properly selecting
the materials of the layers 102, 104 and their relative thickness and
spacing, the ICE 100 will be configured to selectively transmit or
reflect predetermined fractions of electromagnetic radiation at different
wavelengths. Each wavelength is given a predetermined weighting or
loading factor. The thickness and spacing of the layers 102, 104 may be
determined using a variety of approximation methods from the spectrum of
the characteristic or analyte of interest. These methods may include
inverse Fourier transform (IFT) of the optical transmission spectrum and
structuring the ICE 100 as the physical representation of the IFT. The
approximations convert the IFT into a structure based on known materials
with constant refractive indices.

[0023] The weightings that the layers 102, 104 of the ICE 100 apply at
each wavelength are set to the regression weightings described with
respect to a known equation, or data, or spectral signature. For
instance, when electromagnetic radiation interacts with a substance,
unique physical and chemical information about the substance is encoded
in the electromagnetic radiation that is reflected from, transmitted
through, or radiated from the substance. This information is often
referred to as the spectral "fingerprint" of the substance. The ICE 100
is configured to perform the dot product of the received electromagnetic
radiation and the wavelength dependent transmission function (spectrum)
of the ICE 100. The wavelength dependent transmission function of the ICE
100 is dependent on the substrate 106, the material refractive index of
each layer 102, 104, the number of layers 102, 104 and thickness of each
layer 102, 104. As a result, the output light intensity of the ICE 100 is
related to the characteristic or analyte of interest.

[0024] As further explanation, accurately determining the regression
vector of the characteristic of interest in the sample substance provides
a means for an optical computing device to determine or otherwise
calculate a concentration of said characteristic in the sample substance.
The regression vector for each characteristic may be determined using
standard procedures that will be familiar to one having ordinary skill in
the art. For example, analyzing the spectrum of the sample substance may
include determining a dot product of the regression vector for each
characteristic of the sample substance being analyzed. As one of ordinary
skill will recognize, a dot product of a vector is a scalar quantity
(i.e., a real `n` number). While the dot product value is believed to
have no physical meaning by itself (e.g., it may return a positive or
negative result of any magnitude), comparison of the dot product value of
a sample substance with dot product values obtained for known reference
standards and plotted in a calibration curve allows the dot product value
of the sample substance to be correlated with a concentration or value of
a desired characteristic, thereby allowing unknown sample substances to
be accurately analyzed.

[0025] To determine the dot product, one multiples the regression
coefficient of the regression vector at a given wavelength by the
spectral intensity at the same wavelength. This process is repeated for
all wavelengths analyzed, and the products are summed over the entire
wavelength range to yield the dot product. Two or more characteristics
may be determined from a single spectrum of the sample substance by
applying a corresponding regression vector for each characteristic.

[0026] In practice, it is possible to derive information from
electromagnetic radiation interacting with a sample substance by, for
example, separating the electromagnetic radiation from several samples
into wavelength bands and performing a multiple linear regression of the
band intensity against a characteristic of interest determined by another
measurement technique for each sample substance. The measured
characteristic may be expressed and modeled by multiple linear regression
techniques that will be familiar to one having ordinary skill in the art.
Specifically, if y is the measured value of the concentration or
characteristic, y may be expressed as in Equation 1:

[0027] where each `a` is a constant determined by the regression analysis
and each `w` is the light intensity for each wavelength band. Depending
on the circumstances, the estimate obtained from Equation (1) may be
inaccurate, for example, due to the presence of other characteristics
within the sample substance that may affect the intensity of the
wavelength bands. A more accurate estimate may be obtained by expressing
the electromagnetic radiation in terms of its principal components.

[0028] To obtain the principal components, spectroscopic data is collected
for a variety of similar sample substances using the same type of
electromagnetic radiation. For example, following exposure to each sample
substance, the electromagnetic radiation may be collected and the
spectral intensity at each wavelength may be measured for each sample
substance. This data may then be pooled and subjected to a
linear-algebraic process known as singular value decomposition (SVD) in
order to determine the principal components. Use of SVD in principal
component analysis will be well understood by one having ordinary skill
in the art. Briefly, however, principal component analysis is a dimension
reduction technique that takes `m` spectra with `n` independent variables
and constructs a new set of eigenvectors that are linear combinations of
the original variables. The eigenvectors may be considered a new set of
plotting axes. The primary axis, termed the first principal component, is
the vector that describes most of the data variability. Subsequent
principal components describe successively less sample variability, until
the higher order principal components essentially describe only spectral
noise.

[0029] Typically, the principal components are determined as normalized
vectors. Thus, each component of an electromagnetic radiation sample may
be expressed as x.sub.nz.sub.n, where x, is a scalar multiplier and
z.sub.n is the normalized component vector for the n.sup.th component.
That is, z.sub.n is a vector in a multi-dimensional space where each
wavelength is a dimension. Normalization determines values for a
component at each wavelength so that the component maintains its shape
and the length of the principal component vector is equal to one. Thus,
each normalized component vector has a shape and a magnitude so that the
components may be used as the basic building blocks of any
electromagnetic radiation sample having those principal components.
Accordingly, each electromagnetic radiation sample may be described by a
combination of the normalized principal components multiplied by the
appropriate scalar multipliers, as set forth in Equation (2):

x.sub.1z.sub.1+x.sub.2z.sub.2+ . . . +x.sub.nz.sub.n Equation (2)

[0030] The scalar multipliers x.sub.n may be considered the "magnitudes"
of the principal components in a given electromagnetic radiation sample
when the principal components are understood to have a standardized
magnitude as provided by normalization.

[0031] Because the principal components are orthogonal, they may be used
in a relatively straightforward mathematical procedure to decompose an
electromagnetic radiation sample into the component magnitudes, which may
accurately describe the data in the original electromagnetic radiation
sample. Since the original electromagnetic radiation sample may also be
considered a vector in the multi-dimensional wavelength space, the dot
product of the original signal vector with a principal component vector
is the magnitude of the original signal in the direction of the
normalized component vector. That is, it is the magnitude of the
normalized principal component present in the original signal. This is
analogous to breaking a vector in a three dimensional Cartesian space
into its X, Y and Z components. The dot product of the three-dimensional
vector with each axis vector, assuming each axis vector has a magnitude
of 1, gives the magnitude of the three dimensional vector in each of the
three directions. The dot product of the original signal and some other
vector that is not perpendicular to the other three dimensions provides
redundant data, since this magnitude is already contributed by two or
more of the orthogonal axes.

[0032] Moreover, because the principal components are orthogonal to each
other, the dot product of any principal component with any other
principal component is zero. Physically, this means that the components
do not spectrally interfere with each other. If data is altered to change
the magnitude of one component in the original electromagnetic radiation
signal, the other components remain unchanged. In the analogous Cartesian
example, reduction of the X component of the three dimensional vector
does not affect the magnitudes of the Y and Z components.

[0033] Principal component analysis provides the fewest orthogonal
components that can accurately describe the data carried by the
electromagnetic radiation samples. Thus, in a mathematical sense, the
principal components are components of the original electromagnetic
radiation that do not interfere with each other and that represent the
most compact description of the spectral signal. Physically, each
principal component is an electromagnetic radiation signal that forms a
part of the original electromagnetic radiation signal. Each principal
component has a shape over some wavelength range within the original
wavelength range. Summing the principal components may produce the
original signal, provided each component has the proper magnitude,
whether positive or negative.

[0034] The principal components may comprise a compression of the
information carried by the total light signal. In a physical sense, the
shape and wavelength range of the principal components describe what
information is in the total electromagnetic radiation signal, and the
magnitude of each component describes how much of that information is
present. If several electromagnetic radiation samples contain the same
types of information, but in differing amounts, then a single set of
principal components may be used to describe (except for noise) each
electromagnetic radiation sample by applying appropriate magnitudes to
the components. The principal components may be used to provide an
estimate of the characteristic of the sample substance based upon the
information carried by the electromagnetic radiation that has interacted
with that sample substance. Differences observed in spectra of sample
substances having varying quantities of an analyte or values of a
characteristic may be described as differences in the magnitudes of the
principal components.

[0035] Thus, the concentration of the characteristic may be expressed by
the principal components according to Equation (3) in the case where four
principal components are used:

[0036] where `y` is a concentration or value of a characteristic, each a
is a constant determined by the regression analysis, and x.sub.1,
x.sub.2, x.sub.3 and x.sub.4 are the first, second, third, and fourth
principal component magnitudes, respectively. Equation (3) may be
referred to as a regression vector. The regression vector may be used to
provide an estimate for the concentration or value of the characteristic
for an unknown sample.

[0037] Regression vector calculations may be performed by a computer based
on spectrograph measurements of electromagnetic radiation by wavelength.
The spectrograph system spreads the electromagnetic radiation into its
spectrum and measures the spectral intensity at each wavelength over the
wavelength range. Using Equation (3), the computer reads the intensity
data and decomposes the electromagnetic radiation sample into the
principal component magnitudes x.sub.n by determining the dot product of
the total signal with each component. The component magnitudes are then
applied to the regression equation to determine a concentration or value
of the characteristic.

[0038] To simplify the foregoing procedure, however, the regression vector
can be converted to a form that is a function of wavelength so that only
one dot product is determined. Each normalized principal component vector
z.sub.n has a value over all or part of the total wavelength range. If
each wavelength value of each component vector is multiplied by the
regression constant and corresponding to the component vector, and if the
resulting weighted principal components are summed by wavelength, the
regression vector takes the form of Equation (4):

[0039] where a.sub.0 is the first regression constant from Equation (3),
b.sub.n is the sum of the multiple of each regression constant a.sub.n
from Equation (3) and the value of its respective normalized regression
vector at wavelength `n`, and u.sub.n is the intensity of the
electromagnetic radiation at wavelength `n`. Thus, the new constants
define a vector in wavelength space that directly describes a
concentration or characteristic of a sample substance. The regression
vector in the form of Equation (4) represents the dot product of an
electromagnetic radiation sample with this vector.

[0040] Normalization of the principal components provides the components
with an arbitrary value for use during the regression analysis.
Accordingly, it is very unlikely that the dot product value produced by
the regression vector will be equal to the actual concentration or
characteristic value of a sample substance being analyzed. The dot
product result is, however, related (e.g., proportional or having a
logarithmic or exponential relationship) to the concentration or
characteristic value. As discussed above, the relationship may be
determined by measuring one or more known calibration samples by
conventional means and comparing the result to the dot product value of
the regression vector. Thereafter, the dot product result can be compared
to the value obtained from the calibration standards in order to
determine the concentration or characteristic of an unknown sample being
analyzed.

[0041] Before an ICE (i.e., the ICE 100 of FIG. 1) is physically
fabricated for use, the ICE must be designed to be predictive of a
desired characteristic or analyte of interest of a substance being
analyzed. One methodology used to design and fabricate an ICE is commonly
referred to as the "forward design process." In the forward design
process, several (e.g., 100,000+) random designs of the ICE are generated
using a computer-based software program or "design suite." The design
suite is stored on a computer-readable medium containing program
instructions configured to be executed by one or more processors of a
computer system. Ultimately, the design suite generates a theoretical ICE
design for each of the 100,000+ random designs, each being optimized and
otherwise configured to detect a particular characteristic of interest of
a substance desired to be analyzed.

[0042] The design suite commences the design process by generating a
single random ICE design having a random number of layers (i.e., layers
102, 104 of FIG. 1) and/or a random thickness for each layer. The
resulting random ICE design yields a given transmission spectrum
(function), which is intended to match or closely mimic a regression
vector having a series of peaks and valleys that are co-related to the
desired characteristic of interest. The performance of this random ICE
design is then determined by calculating one or more performance criteria
(alternately referred to as "figures of merit") associated with the
random ICE design as compared with a known or measured analyte
concentration of an optical data set corresponding to the desired
characteristic of interest. Example performance criteria include, but are
not limited to, standard error of calibration (SEC), standard error of
prediction (SEP), calibration sensitivity, transmission throughput,
minimum prediction error, slope of the calibration curve, signal-to-noise
ratio, environmental performance characteristics, predictive
concentration range, linearity of prediction, thin-film stack thickness,
individual layer thicknesses, mean transmission value, variability of the
above performance criteria as a function of temperature or fabrication
tolerance, corresponding to the particular characteristic of interest.

[0043] In some forward design processes, the performance of the random ICE
design may be based on its calculated SEC (alternately referred to as
"accuracy"), which is indicative of how predictive the particular ICE
will be for the characteristic of interest during use. The SEC is
generally calculated from a set of test data obtained through the
projected transmission spectrum (function) of the random ICE design and
comparing a predicted result of the characteristic of interest for each
sample in the test set to that of a known value for the characteristic of
interest. It should be noted, however, that the sensitivity of the random
ICE design may equally be calculated from the encoded ICE regression
vector and evaluated for predictability. Sensitivity can be determined by
determining the detector response for the given ICE transmission function
and then plotting this detector response vs. the analyte concentration.
The slope of this plot determines the sensitivity, and the ICE is
designed with the aim of maximizing this slope.

[0044] The design suite then proceeds to iteratively modify the initial
random ICE designs in an attempt to change the transmission function and
thereby improve one or more of the performance criteria. Such
modifications of the random ICE designs that result in changes to the
transmission function include varying layer optical thicknesses and/or
adding or removing layers to the thin film stack. Such iterations are
typically small or minute changes to the random ICE design, such as
altering the thickness of a single layer by as little as 0.01 nanometers
(nm). The result is the generation of a theoretical ICE design that
approaches one or more minimum performance criteria for predicting the
characteristic of interest. The design suite repeats this process of
optimizing the random ICE designs to produce tens of thousands of
theoretical designs. In some cases, the design suite may end up producing
100,000+ theoretical ICE designs from each original random ICE design.

[0045] Once these optimized (theoretical) ICE designs are generated, they
are then sorted by the design suite based on the various performance
criteria described above, such as prediction error and signal. In some
cases, the theoretical ICE designs may be sorted based on their overall
SEC (accuracy) as tested against a known value for the characteristic of
interest. The SEC for each theoretical ICE design may be calculated by
taking the square root of the sum of squares between the known value for
the characteristic of interest and the predicted value as derived from
the transmission function of the particular theoretical ICE design. This
is accomplished for each theoretical ICE design by calculating its
respective transmission function and applying that transmission function
to the known data set of the analyte of interest.

[0046] In some cases, the design suite may be configured to iterate and/or
optimize layer thicknesses and numbers until reaching a reasonable SEC
for one or more of the theoretical ICE designs. In some embodiments, ICE
designs exhibiting an SEC of 2.00 or less, for example, may be considered
"predictive" or "viable" and ICE designs exhibiting an SEC of greater
than 2.00 may be considered "non-predictive." Those ICE designs that are
ultimately considered non-predictive may be removed from consideration
either by an operator or by software instructions carried out by the
design suite.

[0047] Once a predictive ICE design is ultimately selected for fabrication
from the theoretical ICE designs, the predictive ICE design may then be
loaded into a fabrication computer program configured to instruct an
associated fabrication machine or module to physically manufacture the
ICE. The fabrication computer program may be configured to receive and/or
download the specifications for the predictive ICE design from the design
suite and instruct the fabrication machine to physically create a
corresponding ICE by methodically or sequentially depositing the various
layers of the ICE to the specified layer thicknesses.

[0048] Since the forward design process starts with an extremely large set
of random ICE designs (e.g., 100,000+ designs), and each random ICE
design is iteratively optimized as described above, it requires immense
computational capacity and time to generate predictive ICE designs
suitable for fabrication. Moreover, the forward design process can
produce several optimized ICE designs that are substantially identical,
thereby resulting in wasted calculation time for non-unique ICE designs.

[0049] The present disclosure provides novel methods for the design and
fabrication of an ICE. More particularly, the exemplary methods described
herein employ a "reverse design process" in which an ICE transmission
function (spectrum) is optimized irrespective of any thin film stack
parameters (i.e., number of layers, layer thickness, etc.). This offers a
distinctive advantage over the forward design process by allowing the ICE
design to evolve without becoming trapped in a local minimum constrained
by Fresnel equations and/or fabrication conditions that affect the
material properties of a fabricated ICE.

[0050] FIG. 2 is a schematic flowchart of an exemplary method 200 of
designing an ICE using a reverse design process, according to one or more
embodiments. The resulting predictive ICE designs derived through the
method 200 may be similar in some respects to the ICE 100 of FIG. 1, but
may be generated faster and computationally less expensive as compared to
ICEs designed using the forward design process. As illustrated, the
method 200 may first include generating an array of discrete data points
and plotting the discrete data points across a predetermined wavelength
region, as at 202. The predetermined wavelength region may correspond to
a wavelength range where a desired characteristic of interest may be
detected (measured), and the magnitude of each discrete data point may be
randomly or selectively assigned a transmittance value ranging between
zero and 1.

[0051] FIG. 3 is an example transmittance versus wavelength plot 300
showing an array of discrete data points 302 plotted along a
predetermined wavelength region ranging from 2600 nm to 3300 nm. The plot
300 may be generated as a result of step 202 of the method 200 of FIG. 2.
The wavelength region 2600-3300 nm in the plot 300 may correspond to, for
example, the wavelength region where carbon dioxide (CO.sub.2) is
detectable or measureable. Each discrete data point 302 is assigned
(randomly or selectively) a magnitude constrained between zero and 1 that
corresponds to a transmittance value (percentage) for the particular data
point. It should be noted that the array of discrete data points 302 do
not represent any particular ICE design or ICE application, but is
provided for illustrative purposes only in describing the aspects of the
present disclosure.

[0052] The array of discrete data points 302 may be generated using a
computer-operated random number generator. The random number generator
may be operated by MATLAB.RTM. or a similar software program having
instructions capable of being executed by a computer. The random number
generator may be programmed to select a finite number of discrete data
points 302 and randomly project (place) the data points 302 within the
predetermined wavelength region and constrained to a value between zero
and 1. While it is theoretically possible to select and randomly project
hundreds or even thousands of discrete data points 302, it may be
infeasible or inefficient to do so. Accordingly, the random number
generator may be programmed with an upper limit parameter to the number
of allowable data points 302. Theoretically, the upper limit for the
total number of data points 302 may be any number, but would include at
least one data point 302. Accordingly, the random number generator may be
configured to limit the number of allowable data points 302 to a range
between 1 and n-1, where n is the number of wavelength data points. In
the illustrated embodiment, for example, eight data points 302 were
generated and were plotted at 100 nm increments from each other and
randomly assigned a transmittance magnitude ranging between zero and 1.

[0053] In other embodiments, however, the array of discrete data points
302 may be determined based on critical point values for a pre-determined
regression vector corresponding to a character of interest. The critical
point values (extrema) may be obtained by identifying one or more extrema
of the predetermined regression vector corresponding to the
characteristic of interest. The resulting data points may then be
unequally spaced over the predetermined wavelength region and assigned a
transmittance magnitude ranging between zero and 1. The predetermined
regression vector may be determined from a multivariate regression model,
such as partial least squares (PLS) or principal components regression
(PCR).

[0054] Referring again to FIG. 2, the method 200 may then proceed by
generating a line shape that connects to and is constrained by the array
of discrete data points, and thereby generating a first transmission
function, as at 204. The first transmission function extends over the
predetermined wavelength region and is constrained by the values of the
random data points 302 (FIG. 3). FIG. 4 depicts an example first
transmission function 402 fitted to the array of discrete data points 302
initially projected onto the plot 300, as per 202. In some embodiments,
as illustrated, the first transmission function 402 may comprise a
polynomial line function. In other embodiments, however, the first
transmission function 402 may comprise a square or linear line function,
without departing from the scope of the disclosure.

[0055] The first transmission function 402 may be generated using a
computer-operated point-by-point line interpolant process. In some
embodiments, the point-by-point line interpolant process may comprise a
spline function, such as a basis spline (B-spline) function, which may be
operated using MATLAB.RTM. or a similar software program having
instructions capable of being executed by a computer. The B-spline
function provides a piecewise polynomial form of a cubic spline
interpolant to the values of each data point 302 at the corresponding
data sites.

[0056] Referring again to FIG. 2, the method 200 may then proceed by
iteratively modifying the discrete data points of the first transmission
function based on one or more performance criteria to generate a second
transmission function, as at 206. As indicated above, example performance
criteria can include, but are not limited to, SEC (accuracy), standard
error of prediction (SEP), calibration sensitivity, transmission
throughput, minimum prediction error, slope of the calibration curve,
signal-to-noise ratio, environmental performance characteristics,
predictive concentration range, linearity of prediction, thin-film stack
thickness, individual layer thicknesses, mean transmission value, and
variability of the above performance criteria as a function of
temperature or fabrication tolerance.

[0057] Modifying the discrete data points of the first transmission
function based on performance criteria may include iteratively altering
the transmittance value (magnitude) of each discrete data point to
optimize the performance criteria in view of a known or measured analyte
concentration for a desired characteristic of interest. Also, modifying
the discrete data points of the first transmission function based on
performance criteria may include iteratively altering the location of
each discrete data point along the predetermined wavelength region to
optimize the performance criteria in view of the known analyte
concentration of the desired characteristic of interest. In addition,
modifying the discrete data points of the first transmission function
based on performance criteria may include iteratively altering both the
magnitude of the transmittance and the location of each discrete data
point along the predetermined wavelength region.

[0058] FIG. 5 depicts the plot 300 showing the first transmission function
402 and a second or "target" transmission function 502 that results from
modifying the discrete data points 302 based on one or more performance
criteria, as at 206 (FIG. 2). Modifying the discrete data points 302
results in the generation of an array of new discrete data points 504.
Each new discrete data point 504 is an iteration or variation of a
corresponding one of the first discrete data points 302 and is based on
the optimization (i.e., minimization or maximization) of one or more
performance criteria. For instance, the discrete data points 302 of the
first transmission function 402 may be modified by minimizing the SEC
(accuracy) in view of a known or measured analyte concentration of a
desired characteristic of interest. Also, the discrete data points 302 of
the first transmission function 402 may be modified by maximizing the
sensitivity in view of a known or measured analyte concentration of the
desired characteristic of interest over the given analyte concentration
range.

[0059] Once the discrete data points 302 are iteratively modified to
provide the new discrete data points 504, a line shape may be generated
that connects to and is constrained by the array of new discrete data
points 504, and thereby generating the second transmission function 502.
As with the first transmission function 302, the second transmission
function 502 may be generated using a computer-operated point-by-point
line interpolant process, such as a B-spline function. The second
transmission function 502 may then be projected against an optical data
set corresponding to a known or measured analyte concentration of the
characteristic of interest such that the performance criteria may again
be measured. This process may iteratively repeat until the performance
criteria of the new discrete data points 504 and the associated second
transmission function 502 reach a predetermined threshold. In some cases,
for example, the process may iteratively repeat until the new discrete
data points 504 and the associated second transmission function 502 reach
a reasonable SEC or sensitivity threshold.

[0060] In cases where SEC is calculated, a second transmission function
502 exhibiting an SEC of 2.00 or less, for example, may be considered
"predictive" or "viable" and a second transmission function 502
exhibiting an SEC of greater than 2.00 may be considered
"non-predictive." The SEC threshold value that determines whether the
second transmission function 502 is considered predictive or
non-predictive, however, may be greater or less than 2.00, without
departing from the scope of the disclosure. Moreover, it will further be
appreciated that any performance criteria mentioned herein may equally
have a corresponding minimum or maximum performance threshold that the
second transmission function 502 may be measured against to determine if
it is predictive or not.

[0061] Referring again to FIG. 2, the method 200 may then proceed by
fitting a model transmission function for a model ICE design to the
second transmission function and thereby identifying a predictive ICE
design for the desired characteristic of interest, as at 208. Commercial
software packages and/or stand-alone software capable of calculating
transmission functions for given thin film stacks can be used to solve
the Fresnel equations required to find a model ICE design having a given
number of layers (i.e., layers 102 and 104 of FIG. 1) and layer
thicknesses that exhibit a transmission function that most closely
matches the optimized second transmission function over the predetermined
wavelength range. In other words, fitting the model transmission function
for the model ICE design to the second transmission function, as at 208,
may comprise undertaking a portion of the forward design process
generally described above. However, since the second transmission
function 502 (FIG. 5) is already known, fitting the model transmission
function for the model ICE design to the second transmission function 502
is much simpler and more efficient since it is not required to search for
both an unknown ICE transmission function and a model ICE design film
stack, as per the true forward design process.

[0062] FIG. 6 is a transmittance versus wavelength plot 600 depicting a
model transmission function 602 for an exemplary model ICE design fitted
against the optimized second transmission function 502 of FIG. 5. Fitting
the model transmission function 602 of the model ICE design to the second
transmission function may require the generation of the model ICE design
using a computer-based software program. The software program commences
the design process for the model ICE design by generating a random ICE
design having a random number of layers (i.e., layers 102, 104 of FIG. 1)
and/or a random thickness for each layer. The resulting random ICE design
yields the model transmission function 602, and the software program may
then iteratively modify the random ICE design in an attempt to alter the
model transmission function 602 so that is more closely aligns with the
second transmission function 502. Such modifications of the random ICE
design includes varying layer optical thicknesses and/or adding or
removing layers to the thin film stack. The software program repeats this
process until the model transmission function 602 matches or closely
matches the second transmission function 502, at which point the model
ICE design may be considered a predictive ICE design for the desired
characteristic of interest.

[0063] Once a model ICE design is determined to be a predictive ICE
design, the model ICE design may be loaded into a fabrication computer
program configured to instruct an associated fabrication machine or
module to physically manufacture a thin film stack corresponding to the
model ICE design. The fabrication computer program may be configured to
receive and/or download the specifications for the model ICE design from
the software program and instruct the fabrication machine to physically
create a corresponding ICE by methodically or sequentially depositing the
various layers of the ICE to the specified layer thicknesses.

[0064] The foregoing reverse design process provided by the method 200 of
FIG. 2 was compared against the forward design process described above
and the resulting predictive ICE designs from each design methodology
were compared. Table 1 below, for example, shows a comparison of
predictive ICE designs generated through the forward design and reverse
design processes for six characteristics of interest: gas-to-oil ratio
(GOR), methane (C.sub.1), ethane (C.sub.2), propane (C.sub.3), saturates,
and carbon dioxide (CO.sub.2).

[0065] The predictive ICE designs for each design methodology were
optimized with respect to sensitivity. As noted above, sensitivity can be
determined by determining the detector response for the given ICE
transmission function and then plotting this detector response vs. the
analyte concentration. The slope of this plot determines the sensitivity,
and the ICE is designed with the aim of maximizing this slope. For all
six characteristics of interest, the predictive ICE designs generated
using the reverse ICE design process were found to be more sensitive and
were determined (found) significantly faster as compared to the
predictive ICE designs generated using the forward ICE design process.

[0066] The ICEs designed as described herein may be useful in monitoring
or otherwise detecting various analytes or characteristics of substances
related to the oil and gas industry. For instance, the ICEs may be used
in conjunction with an optical computing device to monitor and detect
hydrocarbons, drilling fluids, completion fluids, treatment fluids, etc.
The optical computing devices may be used in a downhole environment, such
as within a wellbore or a tubular extended within the wellbore, or at a
surface location, such as a rig floor, a monitoring facility adjacent a
rig floor, or a remote location where a sample may be delivered for
processing.

[0067] The methods described herein, or large portions thereof, may be
automated at some point such that a computerized system may be programmed
to design, predict, and fabricate ICEs that are more robust for
fluctuating extreme environments. Computer hardware used to implement the
various methods and algorithms described herein can include a processor
configured to execute one or more sequences of instructions, programming
stances, or code stored on a non-transitory, computer-readable medium.
The processor can be, for example, a general purpose microprocessor, a
microcontroller, a digital signal processor, an application specific
integrated circuit, a field programmable gate array, a programmable logic
device, a controller, a state machine, a gated logic, discrete hardware
components, an artificial neural network, or any like suitable entity
that can perform calculations or other manipulations of data. Also,
computer hardware can further include elements such as, for example, a
memory (e.g., random access memory (RAM), flash memory, read only memory
(ROM), programmable read only memory (PROM), electrically erasable
programmable read only memory (EEPROM)), registers, hard disks, removable
disks, CD-ROMS, DVDs, or any other like suitable storage device or
medium.

[0068] Executable sequences described herein can be implemented with one
or more sequences of code contained in a memory. Such code may be read
into the memory from another machine-readable medium. Execution of the
sequences of instructions contained in the memory can cause a processor
to perform the process steps described herein. One or more processors in
a multi-processing arrangement can also be employed to execute
instruction sequences in the memory. In addition, hard-wired circuitry
can be used in place of or in combination with software instructions to
implement various embodiments described herein. Thus, the described
embodiments are not limited to any specific combination of hardware
and/or software.

[0069] As used herein, a machine-readable medium will refer to any medium
that directly or indirectly provides instructions to a processor for
execution. A machine-readable medium can take on many forms including,
for example, non-volatile media, volatile media, and transmission media.
Non-volatile media can include, for example, optical and magnetic disks.
Volatile media can include, for example, dynamic memory. Transmission
media can include, for example, coaxial cables, wire, fiber optics, and
wires that form a bus. Common forms of machine-readable media can
include, for example, floppy disks, flexible disks, hard disks, magnetic
tapes, other like magnetic media, CD-ROMs, DVDs, other like optical
media, punch cards, paper tapes and like physical media with patterned
holes, RAM, ROM, PROM, EPROM and flash EPROM.

[0070] Embodiments disclosed herein include:

[0071] A. A method for designing an integrated computational element (ICE)
that includes generating an array of discrete data points and plotting
the discrete data points across a predetermined wavelength region,
generating a line shape that connects to and is constrained by the array
of discrete data points, and thereby generating a first transmission
function, iteratively modifying the discrete data points based on one or
more performance criteria to generate a second transmission function, and
fitting a model transmission function corresponding to a model ICE design
to the second transmission function and thereby identifying a predictive
ICE design configured to detect a desired characteristic of interest.

[0072] B. A non-transitory, computer readable medium programmed with
computer executable instructions that, when executed by a processor of a
computer unit, perform the method of generating an array of discrete data
points and plotting the discrete data points across a predetermined
wavelength region, generating a line shape that connects to and is
constrained by the array of discrete data points, and thereby generating
a first transmission function, iteratively modifying the discrete data
points based on one or more performance criteria to generate a second
transmission function, and fitting a model transmission function
corresponding to a model ICE design to the second transmission function
and thereby identifying a predictive ICE design configured to detect a
desired characteristic of interest.

[0073] Each of embodiments A and B may have one or more of the following
additional elements in any combination: Element 1: wherein the
predetermined wavelength region corresponds to a wavelength range where
the desired characteristic of interest is detectable. Element 2: wherein
plotting the discrete data points across the predetermined wavelength
region further comprises assigning a transmittance value to each discrete
data point between zero and 1. Element 3: further comprising generating
the array of discrete data points using a computer-operated random number
generator. Element 4: further comprising randomly assigning a
transmittance value to each discrete data point between zero and 1 with
the random number generator. Element 5: wherein generating the array of
discrete data points comprises calculating a predetermined regression
vector corresponding to the characteristic of interest, and selecting
critical point values from the predetermined regression vector, wherein
the critical point values are used as the discrete data points. Element
6: further comprising generating one or both of the first and second
transmission functions using a computer-operated point-by-point line
interpolant process. Element 7: wherein iteratively modifying the
discrete data points based on one or more performance criteria comprises
at least one of determining a standard error of calibration of the second
transmission function in view of the desired characteristic of interest,
and determining an output sensitivity of the second transmission function
in view of the desired characteristic of interest. Element 8: wherein
iteratively modifying the discrete data points comprises at least one of
iteratively altering a transmittance value of each discrete data point to
optimize the one or more performance criteria in view of the desired
characteristic of interest, and iteratively altering a location of each
discrete data point along the predetermined wavelength region to optimize
the one or more performance criteria in view of the desired
characteristic of interest. Element 9: wherein fitting the model
transmission function corresponding to the model ICE design to the second
transmission function comprises generating with a computer the model ICE
design having at least one of a random number of layers and a random
thickness for each layer, iteratively modifying the model ICE design
until the model transmission function aligns with the second transmission
function, and identifying the predictive ICE design once the model
transmission function aligns with the second transmission function.
Element 10: wherein iteratively modifying the model ICE design comprises
at least one of varying the thickness of one or more of the layers and
varying the number of layers. Element 11: further comprising fabricating
an ICE based on the predictive ICE design, and using the ICE in
conjunction with an optical computing device to monitor a substance for a
concentration of the characteristic of interest.

[0074] Element 12: wherein plotting the discrete data points across the
predetermined wavelength region further comprises assigning a
transmittance value to each discrete data point between zero and 1.
Element 13: further comprising generating the array of discrete data
points using a computer-operated random number generator. Element 14:
wherein generating the array of discrete data points comprises
calculating a predetermined regression vector corresponding to the
characteristic of interest, and selecting critical point values from the
predetermined regression vector, wherein the critical point values are
used as the discrete data points. Element 15: further comprising
generating one or both of the first and second transmission functions
using a computer-operated point-by-point line interpolant process.
Element 16: wherein iteratively modifying the discrete data points based
on one or more performance criteria comprises at least one of determining
a standard error of calibration of the second transmission function in
view of the desired characteristic of interest, and determining an output
sensitivity of the second transmission function in view of the desired
characteristic of interest. Element 17: wherein iteratively modifying the
discrete data points comprises at least one of iteratively altering a
transmittance value of each discrete data point to optimize the one or
more performance criteria in view of the desired characteristic of
interest, and iteratively altering a location of each discrete data point
along the predetermined wavelength region to optimize the one or more
performance criteria in view of the desired characteristic of interest.
Element 18: further comprising fabricating an ICE based on the predictive
ICE design.

[0075] By way of non-limiting example, exemplary combinations applicable
to A and B include: Element 3 with Element 4; Element 9 with Element 10;
with Element 11.

[0076] Therefore, the disclosed systems and methods are well adapted to
attain the ends and advantages mentioned as well as those that are
inherent therein. The particular embodiments disclosed above are
illustrative only, as the teachings of the present disclosure may be
modified and practiced in different but equivalent manners apparent to
those skilled in the art having the benefit of the teachings herein.
Furthermore, no limitations are intended to the details of construction
or design herein shown, other than as described in the claims below. It
is therefore evident that the particular illustrative embodiments
disclosed above may be altered, combined, or modified and all such
variations are considered within the scope of the present disclosure. The
systems and methods illustratively disclosed herein may suitably be
practiced in the absence of any element that is not specifically
disclosed herein and/or any optional element disclosed herein. While
compositions and methods are described in terms of "comprising,"
"containing," or "including" various components or steps, the
compositions and methods can also "consist essentially of" or "consist
of" the various components and steps. All numbers and ranges disclosed
above may vary by some amount. Whenever a numerical range with a lower
limit and an upper limit is disclosed, any number and any included range
falling within the range is specifically disclosed. In particular, every
range of values (of the form, "from about a to about b," or,
equivalently, "from approximately a to b," or, equivalently, "from
approximately a-b") disclosed herein is to be understood to set forth
every number and range encompassed within the broader range of values.
Also, the terms in the claims have their plain, ordinary meaning unless
otherwise explicitly and clearly defined by the patentee. Moreover, the
indefinite articles "a" or "an," as used in the claims, are defined
herein to mean one or more than one of the elements that it introduces.
If there is any conflict in the usages of a word or term in this
specification and one or more patent or other documents that may be
incorporated herein by reference, the definitions that are consistent
with this specification should be adopted.

[0077] As used herein, the phrase "at least one of" preceding a series of
items, with the terms "and" or "or" to separate any of the items,
modifies the list as a whole, rather than each member of the list (i.e.,
each item). The phrase "at least one of" allows a meaning that includes
at least one of any one of the items, and/or at least one of any
combination of the items, and/or at least one of each of the items. By
way of example, the phrases "at least one of A, B, and C" or "at least
one of A, B, or C" each refer to only A, only B, or only C; any
combination of A, B, and C; and/or at least one of each of A, B, and C.